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arXiv:1210.3118 · Chinese Physics B

Average position in quantum walks with a U(2) coin

Min Li, Yong-Sheng Zhang, Guang-Can Guo

Published Mar 1, 2013• 10 citations
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Abstract

We investigated discrete-time quantum walks with an arbitary unitary coin. Here we discover that the average position 〈 x 〉 = max(〈 x 〉) sin(α + γ), while the initial state is 1/√2(|0 L ⟩+i|0 R ⟩). We verify the result, and obtain some symmetry properties of quantum walks with a U(2) coin with |0 L 〉 and |0 R 〉 as the initial state.

DOI
10.1088/1674-1056/22/3/030310

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We investigated discrete-time quantum walks with an arbitary unitary coin. Here we discover that the average position 〈 x 〉 = max(〈 x 〉) sin(α + γ), while the initial state is 1/√2(|0 L ⟩+i|0 R ⟩). We verify the result, and obtain some symmetry properties of quantum walks with a U(2) coin with |0 L 〉 and |0 R 〉 as the initial state.
We investigated discrete time quantum walks with an arbitrary initial state ∣Ψ0(θ,ϕ,φ)⟩=cos⁡θeiϕ∣0L⟩+sin⁡θeiφ∣0R⟩\midΨ_{0}(θ,ϕ,φ)\rangle=\cosθe^{iϕ}\mid0L\rangle+\sinθe^{iφ}\mid0R\rangle∣Ψ0​(θ,ϕ,φ)⟩=cosθeiϕ∣0L⟩+sinθeiφ∣0R⟩ with a U(2) coin U(α,β,γ)U(α,β,γ)U(α,β,γ). We discover that the average position xˉ=max⁡(xˉ)cos⁡(α+γ+ϕ−φ)\bar{x}=\max(\bar{x})\cos(α+γ+ϕ-φ)xˉ=max(xˉ)cos(α+γ+ϕ−φ), with coin operator U(α,π/4,γ)U(α,π/4,γ)U(α,π/4,γ) and initial state ∣Φ0(π/4,ϕ,φ)⟩=(eiϕ∣0L⟩+eiφ∣0R⟩)2/2\midΦ_{0}(π/4,ϕ,φ)\rangle=(e^{iϕ}\mid0L\rangle+e^{iφ}\mid0R\rangle)\sqrt{2}/2∣Φ0​(π/4,ϕ,φ)⟩=(eiϕ∣0L⟩+eiφ∣0R⟩)2​/2. If we set initial state and coin operator to ∣Φ0⟩(θ,π/2,0)=icos⁡θ∣0L⟩+sin⁡θ∣0R⟩)\midΦ_{0}\rangle(θ,π/2,0)=i\cosθ\mid0L\rangle+\sinθ\mid0R\rangle)∣Φ0​⟩(θ,π/2,0)=icosθ∣0L⟩+sinθ∣0R⟩) and coin operator U(0,π/4,0)U(0,π/4,0)U(0,π/4,0), for α+γ+ϕ−φ=π/2α+γ+ϕ-φ=π/2α+γ+ϕ−φ=π/2, we discover that xˉ=−max⁡(xˉ)cos⁡(2θ).\bar{x}=-\max(\bar{x})\cos(2θ).xˉ=−max(xˉ)cos(2θ). Last we verify the result above, and obtain the summarize properties of quantum walks with an arbitrary state. We get that xˉ(θ,ϕ,φ,α,β,γ,t)=cos⁡2θ∗xˉ∣0L⟩(β,t)+sin⁡2θ∗cos⁡(α+γ+ϕ−φ)∗xˉ(∣0L⟩+∣0R⟩)2/2(α=γ=0,β,t)\bar{x}(θ,ϕ,φ,α,β,γ,t)=\cos2θ*\bar{x}_{|0L\rangle}(β,t)+\sin2θ*\cos(α+γ+ϕ-φ)*\bar{x}_{(\mid0L\rangle+\mid0R\rangle)\sqrt{2}/2}(α=γ=0,β,t)xˉ(θ,ϕ,φ,α,β,γ,t)=cos2θ∗xˉ∣0L⟩​(β,t)+sin2θ∗cos(α+γ+ϕ−φ)∗xˉ(∣0L⟩+∣0R⟩)2​/2​(α=γ=0,β,t). If the average positions xˉ\bar{x}xˉ with initial state ∣0L⟩|0L\rangle∣0L⟩ and ∣Ψ0⟩=(∣0L⟩+∣0R⟩)2/2\midΨ_{0}\rangle=(\mid0L\rangle+\mid0R\rangle)\sqrt{2}/2∣Ψ0​⟩=(∣0L⟩+∣0R⟩)2​/2 and coin operator U(0,β,0)U(0,β,0)U(0,β,0) are known, we can get the average position result of quantum walks with an arbitrary initial state and a U(2) coin operator.
We investigate the spreading behavior of the one-dimensional discrete time quantum walk starting from local and Gaussian states. In order to compare both cases, we obtain analytically the average variance of the position by averaging over all spin states. We find out that the average variance for the local case is independent on the quantum coin. However, when the quantum walk starts from a Gaussian state, its average variance depends on the sum of relative phases between spin states given by the quantum coin, and also, on the initial dispersion of the state. We perform numerical simulations of the average probability and variance along the time to confront them with our analytical results.
We investigate the ballistic spreading behavior of the one-dimensional discrete time quantum walks whose time evolution is driven by any balanced quantum coin. We obtain closed-form expressions for the long-time variance of position of quantum walks starting from any initial qubit (spin-1/21/21/2 particle) and position states following a delta-like (local), Gaussian and uniform probability distributions. By averaging over all spin states, we find out that the average variance of a quantum walk starting from a local state is independent of the quantum coin, while from Gaussian and uniform states it depends on the sum of relative phases between spin states given by the quantum coin, being non-dispersive for a Fourier walk and large initial dispersion. We also perform numerical simulations of the average probability distribution and variance along the time to compare them with our analytical results.
We explore a discrete-time, coined quantum walk on a quantum network where the coherent superposition of walker moves originates from the unitary interaction of the walker coin with the qubit degrees of freedom in the quantum network. The walk dynamics leads to a growth of entanglement between the walker and the network on one hand and, on the other, between the network qubits among themselves. The initial entanglement among the network qubits plays a crucial role in determining the asymptotic values of these entanglement measures and the quantum walk statistics. Specifically, the entanglement entropy of the walker-network state and the negativity of the quantum network-qubit state saturate to values increasing with the initial network entanglement. The asymptotic time-averaged walker-position probability distribution shows increasing localization around the initial walker position with higher initial network entanglement. A potential application of these results as a characterization tool for quantum network properties is suggested.
For a discrete time quantum walk (QW) on the N-cycle, allowing for decoherence on the coin, we derive a number of results, including an explicit formula for the position probability distribution. For a QW of this type, we show that the mixing behavior tends, in the long run, to a uniform distribution regardless of the initial state of the system and irrespective of the parity of the number of nodes N. These results confirm the findings of previous authors who arrived at similar conclusions through extensive numerical simulations. In particular, we infer that the mixing time M(epsilon) for the time-averaged probability distribution is of order no greater than O(N2/).
A unit evolution step of discrete-time quantum walks (QWs) is determined by both a coin-flip operator and a position-shift operator. The behavior of quantum walkers after many steps delicately depends on the coin-flip operator and an initial condition of the walk. To get the behavior, a lot of long-time limit distributions for the QWs starting with a localized initial state have been derived. In this paper, we compute limit distributions of a 2-state QW with a delocalized initial state, not a localized initial state, and discuss how the walker depends on the coin-flip operator. The initial state induced from the Fourier series expansion, which is called the (α, β) delocalized initial state in this paper, provides different limit density functions from the ones of the quantum walk with a localized initial state.